The retarded Green's function is generally determined by using the Matsubara formalism [1]}. In this
case, we use the imaginary time Green's function, defined by \(\hbar \mathcal {U}(\tau )=-\langle T_\tau \hat{A}(\tau )\hat{A}^\dagger (0)\rangle \) ,
to make the association in the Fourier space \(\mathcal {U}(i\hbar \nu _n)|_{i\hbar \nu _n\rightarrow \delta \mu +i\varepsilon }=U_\textrm {ret}(\delta \mu )\) ,
where \(i\hbar \nu _n= n\pi /\beta \)  are the bosonic (fermionic) frequencies for \(n\)  even (odd).
However, the correspondence provided by the analytic continuation does not work when dealing with coherent states, and the
retarded Green's function must be solved in real time formalism. Therefore, the commutator present is
\(U_\textrm {ret}(t)\)  is given by

rCl
[A0(t),A(0)]cs=2Seffkkq|kkq|2 (fk-fk)ei(k-k)t[qq(q2 eiq t+q2 e-it)+.
